Quantum Fisher Information as a Spectral Entanglement Witness: the Projective Stability Gap and the Origin of Scale-Free Growth

A recent inelastic-neutron-scattering experiment on the heavy-fermion strange metal Ce₃Pd₂₀Si₆ reports a scale-free increase of the quantum Fisher information (QFI) density as the strange-metal state forms, witnessing high multipartite entanglement. We ask whether this signature is reproduced by the non-injective projection framework of Cosmochrony, and we isolate the structural condition under which it is. We first reduce the QFI density to the projective spectral data: using linear response, the QFI density is a weighted integral of the imaginary part of the dynamical susceptibility, whose infrared behaviour is controlled by the projective stability gap Δ_Π and the edge spectral weight of the projective stability operator L_Π. This yields a clean dichotomy. A finite gap forces the QFI density to saturate; a gap that closes with an anomalous edge density of states of exponent a forces fQ(T)~ T¹⁻ᵃ, so that scale-free growth requires $a>1$. We then evaluate the two ingredients by exact diagonalisation. The single winding sector $w=2$ reduces to the critical almost-Mathieu (Harper) operator at flux $2/q$; its gap closes as Δ_Π~ q⁻¹ but its edge spectral dimension is dₛ→ 2, giving a→ 1, the marginal boundary, hence only logarithmic growth. The full Cayley substrate of Heis₃(Z/qZ), whose homogeneous growth dimension is $D=4$, instead has an edge spectral dimension dₛ≈ 4, giving a≈ 2 and fQ(T)~ T⁻¹, a genuine power law. The conclusion is structural: in this framework the scale-free QFI growth is a property of the full nilpotent substrate and not of any isolated composite, consistent with the collective, multipartite character of the measured entanglement. The required thermal closure Δ_Π(T)~ T is obtained at the level of the effective heat-kernel response: the framework identifies the diffusion scale with inverse temperature, and the induced linear-response flow gives Δ_Π(T)=dₛ/2T, so that the chain from the nilpotent substrate to the linear prediction fQ(T)~ T⁻¹ closes within the effective response regime. The predicted exponent is of the right order but steeper than observed. Numerically, the Born–Infeld bounded-response correction can soften the effective exponent from the linear value αₜₕ=1 towards the observed αₑₓₚ≈ 0.7, with a size-stable crossing at a partial substrate saturation. The asymptotic exponent and the saturation level realised by the material remain open.